Difference between open sphere and epsilon-neighbourhood - Metric Spaces

In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows:

An open sphere [itex]S(x_0,r)[/itex] in a metric space [itex]R[/itex] (with metric function [itex]\rho(x,y)[/itex]) is the set of all points [itex]x\in R[/itex] satisfying [itex]\rho(x,x_0)<r[/itex]. The fixed point [itex]x_0[/itex] is called the center; the number [itex]r[/itex] is called the radius.

An ε-neighbourhood of the point [itex]x[/itex], denoted [itex]O(x,\epsilon)[/itex], is an open sphere of radius ε and center [itex]x_0[/itex].

How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point [itex]x[/itex] (as per the definition) does this [itex]x[/itex] have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing?

In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows:

An open sphere [itex]S(x_0,r)[/itex] in a metric space [itex]R[/itex] (with metric function [itex]\rho(x,y)[/itex]) is the set of all points [itex]x\in R[/itex] satisfying [itex]\rho(x,x_0)<r[/itex]. The fixed point [itex]x_0[/itex] is called the center; the number [itex]r[/itex] is called the radius.

An ε-neighbourhood of the point [itex]x[/itex], denoted [itex]O(x,\epsilon)[/itex], is an open sphere of radius ε and center [itex]x_0[/itex].

No. an [itex]\epsilon-neighborhood[/itex] of the point [itex]x[/itex] is open sphere with radius [itex]\epsilon[/itex] and center x, not some [itex]x_0[/itex] (that wouldn't make sense because [itex]x_0[/itex] is not mentioned in the notation "[itex]O(x, \epsilon)[/itex]").

How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point [itex]x[/itex] (as per the definition) does this [itex]x[/itex] have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing?

Thanks.

An "[itex]\epsilon[/itex]-neighborhood of x" is specifically the open ball of radius [itex]\epsilon[/itex] with radius [itex]\epsilon[/itex] that is centered at x.